Brief Summary of the Pulse Synthesis Problem
The synthesis of RF pulse sequences to produce selective excitation is a problem of principal importance in all applications of nuclear magnetic resonance. To discuss this problem, one needs to introduce the Bloch equation without relaxation, which is usually written in the form:
                                          ⅆ            M                                ⅆ            t                          =                  γ          ⁢                                          ⁢                                    M              ⨯              B                        .                                              (        1        )            Here M is the magnetization, B is the applied magnetic field, t is time and γ is the gyromagnetic ratio. A solution to this equation has constant length which is assumed throughout to equal 1. The Bloch equation is usually analyzed in a “rotating reference” frame. Ordinarily the rotating reference frame is related to the “laboratory frame” by a time dependent orthogonal transformation of the form:
                                                        F              ⁡                              (                t                )                                      =                          [                                                                                          cos                      ⁢                                                                                          ⁢                                              θ                        ⁡                                                  (                          t                          )                                                                                                                                                                        -                        sin                                            ⁢                                                                                          ⁢                                              θ                        ⁡                                                  (                          t                          )                                                                                                                          0                                                                                                              sin                      ⁢                                                                                          ⁢                                              θ                        ⁡                                                  (                          t                          )                                                                                                                                                                        cos                        ⁡                                                  (                          θ                          )                                                                    ⁢                                              (                        t                        )                                                                                                  0                                                                                        0                                                        0                                                        1                                                              ]                                ,                                          ⁢                      so            ⁢                                                  ⁢            that                          ⁢                                  ⁢                              M            ⁡                          (              t              )                                =                                    F              ⁡                              (                t                )                                      ⁢                                          m                ⁡                                  (                  t                  )                                            .                                                          (        2        )            One uses m to denote the magnetization in the rotating reference frame. Larmour's Theorem implies that if M satisfies (1) then m satisfies
                                                        ⅆ              m                                      ⅆ              t                                =                      γ            ⁢                                                  ⁢                          m              ⨯                              B                eff                                                    ⁢                                  ⁢        where                            (        3        )                                                      B            eff                    ⁡                      (            t            )                          =                                                                              F                                      -                    1                                                  ⁡                                  (                  t                  )                                            ⁢                              B                ⁡                                  (                  t                  )                                                      +                                          1                γ                            ⁢                              Ω                ⁡                                  (                  t                  )                                            ⁢                                                                                ⁢                                                                              ⁢              with              ⁢                                                          ⁢                              Ω                ⁡                                  (                  t                  )                                                              =                                                    (                                  0                  ,                  0                  ,                                                            θ                      t                                        ⁡                                          (                      t                      )                                                                      )                            t                        .                                              (        4        )            
In most applications of this method, the function θ is selected to render the z-component of Beff independent of time,Beff(f,t)=(f1(t),ω2(t),γ−1f).The constant value f is called the offset frequency or resonance offset. If one sets ω(t)=ω1(t)+iω2(t) then, in the laboratory frame, the RF-envelope takes the form:p(t)=ω(t)efθ(t).The energy in the RF-envelope is given by
                              W          p                =                                            ∫                              -                ∞                            ∞                        ⁢                                                                                                  p                    ⁡                                          (                      t                      )                                                                                        2                            ⁢                                                          ⁢                              ⅆ                t                                              =                                    ∫                              -                ∞                            ∞                        ⁢                                                                                                  ω                    ⁡                                          (                      t                      )                                                                                        2                            ⁢                                                          ⁢                                                ⅆ                  t                                .                                                                        (        6        )            The magnetization profile is a unit 3-vector valued function defined for fε,
            m      ∞        ⁡          (      f      )        =            [                                                                  m                1                ∞                            ⁡                              (                f                )                                                                                                        m                2                ∞                            ⁡                              (                f                )                                                                                                        m                3                ∞                            ⁡                              (                f                )                                                        ]        .  
In essentially all MR applications m∞(f)=[0, 0, 1]t outside of a finite interval. In most earlier approaches to pulse synthesis, the flip angle profile, defined as:φ(f)=cos−1(m3∞(f)),was used to design the pulse, with the phase of transverse magnetization determined, implicitly, by the algorithm. Typically, this is referred to as “recovering” the phase.
The problem of RF-pulse synthesis is to find a time dependent complex pulse envelope ω(t) so that, if Beff(f) is given by (5), then the solution of:
                                                                        ⅆ                m                                            ⅆ                t                                      ⁢                          (                              f                ;                t                            )                                =                                    γ              ⁢                                                          ⁢                                                m                  ⁡                                      (                                          f                      ;                      t                                        )                                                  ⨯                                                      B                                          eff                      ⁢                                                                                                                            ⁡                                      (                                          f                      ;                      t                                        )                                                              ⁢                                                          ⁢              with              ⁢                                                                                ⁢                                                                              ⁢                                                lim                                      t                    →                                          -                      ∞                                                                      ⁢                                  m                  ⁡                                      (                                          f                      ;                      t                                        )                                                                        =                                          (                                  0                  ,                  0                  ,                  1                                )                            t                                      ⁢                                  ⁢        satisfies                            (        7        )                                                      lim                          t              →              ∞                                ⁢                      [                                                                                ⅇ                                                                  -                        t                                            ⁢                                                                                          ⁢                      fI                                                        ⁡                                      (                                                                  m                        1                                            +                                              ⅈ                        ⁢                                                                                                  ⁢                                                  m                          2                                                                                      )                                                  ⁢                                  (                                      f                    ;                    t                                    )                                            ,                                                m                  3                                ⁡                                  (                                      f                    ;                    t                                    )                                                      ]                          =                              [                                                            (                                                            m                      1                      ∞                                        +                                          ⅈ                      ⁢                                                                                          ⁢                                              m                        2                        ∞                                                                              )                                ⁢                                  (                  f                  )                                            ,                                                m                  3                  ∞                                ⁡                                  (                  f                  )                                                      ]                    .                                    (        8        )            The standard complex notation, m1+im2 has been used for the transverse components of the magnetization. If ω1(t)+iω2(t) is supported in the interval [t0, t1], then these asymptotic conditions are replaced by:m(v;t0)=[0,0,1]†,[e−Iwt1(m1+im2)(v;t1),m3(v;t1)]=[(m1∞+im2∞)(v),m3∞(v)].  (9)The mapping from ω1(t)+iω2(t) to m∞ (as defined in Equations (7) and (8)) is highly nonlinear; the problem of pulse synthesis is that of inverting this mapping.
To solve the problem of RF-pulse synthesis, it is convenient to introduce the spin domain formulation of the Bloch equation. Instead of a unit vector m in 3, one solves for a unit vector ψ in 2. This vector satisfies the 2×2 matrix equation:
                                          ⅆ            ψ                                ⅆ            t                          =                              -                          1              2                                ⁢                      ω            ·                          σψ              .                                                          (        10        )            Here ω=−[γω1(t),γω2(t),v], and σ are the Pauli spin matrices:
                                          σ            1                    =                      [                                                            0                                                  1                                                                              1                                                  0                                                      ]                          ,                              σ            2                    =                      [                                                            0                                                                      -                    ⅈ                                                                                                                    -                    ⅈ                                                                    0                                                      ]                          ,                              σ            3                    =                                    [                                                                    1                                                        0                                                                                        0                                                                              -                      1                                                                                  ]                        .                                              (        11        )            Assembling the pieces, it can be seen that ψ satisfies:
                                                                        ⅆ                ψ                                            ⅆ                t                                      ⁢                          (                              ξ                ;                t                            )                                =                                    [                                                                                          -                      ⅈξ                                                                                                  q                      ⁡                                              (                        t                        )                                                                                                                                                        -                                                                        q                          *                                                ⁡                                                  (                          t                          )                                                                                                                                                ⅈ                      ⁢                                                                                          ⁢                      ξ                                                                                  ]                        ⁢            ψ            ⁢                                                  ⁢                          (                              ξ                ;                t                            )                                      ,                                  ⁢        with                            (        12        )                                          ξ          =                      v            2                          ,                              q            ⁡                          (              t              )                                =                                                                      -                  ⅈ                                ⁢                                                                  ⁢                y                            2                        ⁢                                          (                                                                            ω                      1                                        ⁡                                          (                      t                      )                                                        -                                      ⅈ                    ⁢                                                                                  ⁢                                                                  ω                        2                                            ⁡                                              (                        t                        )                                                                                            )                            .                                                          (        13        )            where q* is the complex conjugate of the complex number q.
A simple recipe takes a solution of Equation (12) and produces a solution of Equation (7). If ψ(ξ;t)=[ψ1(ξ;t), ψ2(ξ;t)]† satisfies Equation (12), then the 3-vector:m(v,t)=[2Re(ψ18ψ2),2Im(ψ18ψ2), |ψ1|2-|ψ2|2]†(v/2;t) (14)satisfies Equation (7). If in addition:
                                          lim                          t              →                              -                ∞                                              ⁢                                    ⅇ                              ⅈ                ⁢                                                                  ⁢                ξ                ⁢                                                                  ⁢                t                                      ⁢                          ψ              ⁡                              (                                  ξ                  ;                  t                                )                                                    =                  [                                                    1                                                                    0                                              ]                                    (        15        )            then m satisfies Equation (7). Thus, the RF-pulse synthesis problem is easily translated into an inverse scattering problem for Equation (12). This is described in the next section. Following the standard practice in inverse scattering, Equation (12) is referred to as the ZS-system and q as the potential.Scattering Theory for the ZS-System
Scattering theory for equations like those above relates the behavior of ψ(ξ;t), as t→−∞ to that of ψ(ξ;t); as t→+∞. If q has bounded support, then the functions:
                              [                                                                      ⅇ                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                    ξ                    ⁢                                                                                  ⁢                    t                                                                                                      0                                              ]                ,                  [                                                    0                                                                                      ⅇ                                      ⅈ                    ⁢                                                                                  ⁢                    ξ                    ⁢                                                                                  ⁢                    t                                                                                ]                                    (        16        )            are a basis of solutions outside the support of q. If the L1-norm of q is finite, then, it is known that there are solutions that are asymptotic to these solutions as t→±∞.Theorem 1. If ∥q∥L1 is finite, then, for every real ξ; there are unique solutions:ψ1+(ξ), ψ2+(ξ) and ψ1−(ξ), ψ2−(ξ)  (17)which satisfy:
                                                        lim                              t                →                                  -                  ∞                                                      ⁢                                          ⅇ                                  ⅈξ                  ⁢                                                                          ⁢                  t                                            ⁢                                                ψ                                      1                    -                                                  ⁡                                  (                                      ξ                    ;                    t                                    )                                                              =                      [                                                            1                                                                              0                                                      ]                          ,                                            lim                              t                →                                  -                  ∞                                                      ⁢                                          ⅇ                                                      -                    ⅈξ                                    ⁢                                                                          ⁢                  t                                            ⁢                                                ψ                                      2                    -                                                  ⁡                                  (                                      ξ                    ;                    t                                    )                                                              =                      [                                                            0                                                                                                  -                    1                                                                        ]                                              (        18        )                                                                    lim                              t                →                ∞                                      ⁢                                          ⅇ                                  ⅈξ                  ⁢                                                                          ⁢                  t                                            ⁢                                                ψ                                      1                    +                                                  ⁡                                  (                                      ξ                    ;                    t                                    )                                                              =                      [                                                            1                                                                              0                                                      ]                          ,                                            lim                              t                →                ∞                                      ⁢                                          ⅇ                                                      -                    ⅈξ                                    ⁢                                                                          ⁢                  t                                            ⁢                                                ψ                                      2                    +                                                  ⁡                                  (                                      ξ                    ;                    t                                    )                                                              =                      [                                                            0                                                                              1                                                      ]                                              (        19        )            The solutions ψ1−(ξ), ψ2+(ξ) extend as analytic functions of ξ to the upper half plane, {ξ: Im >0} and ψ2−(ξ), ψ1+(ξ) extend as analytic functions of ξ to the lower half plane, {ξ: Imξ>0}. The proof of this theorem can be found in Ablowitz, et al., “The inverse scattering transform-Fourier analysis for nonlinear problems,” Studies in Applied Math., Vol. 53, (1974), pp. 249-315.
For real values of ξ the solutions normalized at −∞ can be expressed in terms of the solutions normalized at +∞ by linear relations:ψ1−(ξ;t)=a(ξ)ψ1+(ξ;t)+b(ξ)ψ2+(ξ;t),ψ2−(ξ;t)=b*(ξ)ψ1+(ξ;t)−a*(ξ)ψ2+(ξ;t).  (20)The functions a; b are called the scattering coefficients for the potential q. The 2×2-matrices [ψ1−ψ2−], [ψ1+ψ2+] satisfy:
                              [                                    ψ              1                        -                          ψ                              2                -                                              ]                =                                            [                                                ψ                                      1                    +                                                  ⁢                                  ψ                                      2                    +                                                              ]                        ⁡                          [                                                                                          a                      ⁡                                              (                        ξ                        )                                                                                                                                                b                        *                                            ⁡                                              (                        ξ                        )                                                                                                                                                        b                      ⁡                                              (                        ξ                        )                                                                                                                        -                                                                        a                          *                                                ⁡                                                  (                          ξ                          )                                                                                                                                ]                                .                                    (        21        )            The scattering matrix for the potential q is defined to be:
                              s          ⁡                      (            ξ            )                          =                              [                                                                                a                    ⁡                                          (                      ξ                      )                                                                                                                                  b                      *                                        ⁡                                          (                      ξ                      )                                                                                                                                        b                    ⁡                                          (                      ξ                      )                                                                                                            -                                                                  a                        *                                            ⁡                                              (                        ξ                        )                                                                                                                  ]                    .                                    (        22        )            It is not difficult to show that:a(ξ)=[ψ11−(ξ;t)ψ22+(ξ;t)−ψ21−(ξ;t)ψ12+(ξ;t)]  (23)It follows from Theorem 1 and Equation (23) that a extends to the upper half plane as an analytic function. Also, if {ξ1, . . . , ξN} is a list of the zeros of a, then Equation (23) implies that for each j there is a nonzero complex number C′j so that:ψ1−(ξj)=C′jψ2+(ξj), j=1, . . . , N. (24)
The reflection coefficient is defined by:
                              r          ⁡                      (            ξ            )                          =                                            b              ⁡                              (                ξ                )                                                    a              ⁡                              (                ξ                )                                              .                                    (        25        )            A priori the reflection coefficient is only defined on the real axis. Since |a(ξ)|2+|b(ξ)|2=1, a(ξ) may be expressed in terms of r(ξ) as:
                                          a            ⁡                          (              ξ              )                                =                                    ∏                              j                =                1                            n                        ⁢                                          (                                                      ξ                    -                                                                  ξ                        ⁢                                                                                                                      j                                                                            ξ                    -                                          ξ                      j                      *                                                                      )                            ⁢                                                          ⁢                                                exp                  ⁡                                      [                                                                  ⅈ                                                  2                          ⁢                          π                                                                    ⁢                                                                        ∫                                                      -                            ∞                                                    ∞                                                ⁢                                                                                                            log                              ⁡                                                              (                                                                  1                                  +                                                                                                                                                                                          r                                        ⁡                                                                                  (                                          ζ                                          )                                                                                                                                                                                            2                                                                                                  )                                                                                      ⁢                                                          ⅆ                              ζ                                                                                                            ζ                            -                            ξ                                                                                                                ]                                                  .                                                    ⁢                                                      (        26        )            Equation (26) has a well defined limit as ξ approaches the real axis.
If a has simple zeros at the points {ξ1, . . . , ξN} (so that a′(ξj)≠0), then the norming constants are defined by setting:
                                          c            j                    =                                    c              j              ′                                                      a                ′                            ⁡                              (                                  ξ                  j                                )                                                    ,                            (        27        )            where the {C′j} are defined in Equation (24). The definition needs to be modified if a has nonsimple zeros. The reason for replacing {C′j} with {Cj} win become more apparent below. The pairs {(ξj,Cj)} are often referred to as the discrete data, or bound state data.
The function r(ξ), for ξε and the collection of pairs {(ξj,Cj):j=1, . . . , N} define the reduced scattering data. Implicitly the reduced scattering data is a function of the potential q. In inverse scattering theory, the data {r(ξ) for ξε; (ξ1,C1), . . . ,(εN,CN) are specified, and a potential q is sought that has this reduced scattering data. The map from the reduced scattering data, or the mathematically equivalent data, to q is often called the Inverse Scattering Transform or IST.
The RF-pulse synthesis problem will now be rephrased as an inverse scattering problem. Since the data for the pulse synthesis problem is the magnetization profile m∞, a function of ξ=v/2. The solution ψ1− to the ZS-system defines a solution m1− to:
                                                                        ⅆ                m                                            ⅆ                t                                      ⁢                          (                              v                ;                t                            )                                =                      γ            ⁢                                                  ⁢                                          m                ⁡                                  (                                      v                    ;                    t                                    )                                            ⨯                                                B                  eff                                ⁡                                  (                                      v                    ;                    t                                    )                                                                    ⁢                                  ⁢        with                            (        28        )                                                                    lim                              t                →                                  -                  ∞                                                      ⁢                          m              ⁡                              (                                  v                  ;                  t                                )                                              =                                    [                              0                ,                0                ,                1                            ]                        +                          ,                            (        29        )            It follows from Equations (19) and (20) that:
                                                        ψ                              1                -                                      ⁡                          (                              ξ                ;                t                            )                                ~                      [                                                                                                      a                      ⁡                                              (                        ξ                        )                                                              ⁢                                          ⅇ                                                                        -                          ⅈ                                                ⁢                                                                                                  ⁢                        ξ                        ⁢                                                                                                  ⁢                        t                                                                                                                                                                                    b                      ⁡                                              (                        ξ                        )                                                              ⁢                                          ⅇ                                              ⅈ                        ⁢                                                                                                  ⁢                        ξ                        ⁢                                                                                                  ⁢                        t                                                                                                                  ]                          ,                              as            ⁢                                                  ⁢            t                    →                                    +                              ∞                .                                                                  ⁢                Therefore                                      ⁢                          :                                                          (        30        )                                                                    m                              1                -                                      ⁡                          (                              ξ                ;                t                            )                                ~                      [                                                                                2                    ⁢                                          b                      ⁡                                              (                        ξ                        )                                                              ⁢                                                                  a                        *                                            ⁡                                              (                        ξ                        )                                                              ⁢                                          ⅇ                                              2                        ⁢                        ⅈ                        ⁢                                                                                                  ⁢                        ξ                        ⁢                                                                                                  ⁢                        t                                                                                                                                                                                                                                                              a                          ⁡                                                      (                            ξ                            )                                                                                                                      2                                        -                                                                                                                    b                          ⁡                                                      (                            ξ                            )                                                                                                                      2                                                                                            ]                          ,                              as            ⁢                                                  ⁢            t                    →                      +                          ∞              .                                                          (        31        )            
The complex notation for the transverse components of m1− are used. If m1− also satisfies:
                                                        lim                              t                →                ∞                                      ⁢                          [                                                                                          ⅇ                                                                        -                          ⅈ                                                ⁢                                                                                                  ⁢                        vt                                                              ⁡                                          (                                                                        m                          1                                                +                                                  ⅈ                          ⁢                                                                                                          ⁢                                                      m                            2                                                                                              )                                                        ⁢                                      (                                          v                      ;                      t                                        )                                                  ,                                                      m                    3                                    ⁡                                      (                                          v                      ;                      t                                        )                                                              ]                                =                      [                                                            (                                                            m                      1                      ∞                                        +                                          ⅈ                      ⁢                                                                                          ⁢                                              m                        2                        ∞                                                                              )                                ⁢                                  (                  v                  )                                            ,                                                m                  3                  ∞                                ⁡                                  (                  v                  )                                                      ]                          ,                            (        32        )            then it follows from Equation (31) and |a(ξ)|2+|b(ξ)|2=1 that:
                                                                        r                ⁡                                  (                  ξ                  )                                            =                                                                    b                    ⁡                                          (                      ξ                      )                                                                            a                    ⁡                                          (                      ξ                      )                                                                      =                                                      lim                                          t                      →                      ∞                                                        ⁢                                                                                    (                                                                              m                                                          11                              -                                                                                +                                                      ⅈ                            ⁢                                                                                                                  ⁢                                                          m                                                              21                                -                                                                                                                                    )                                            ⁢                                              (                                                  ξ                          ;                          t                                                )                                            ⁢                                              ⅇ                                                                              -                            2                                                    ⁢                          ⅈξt                                                                                                            1                      +                                                                        m                                                      31                            -                                                                          ⁡                                                  (                                                      ξ                            ;                            t                                                    )                                                                                                                                                                                            =                                                                                          (                                                                        m                          1                          ∞                                                +                                                  ⅈ                          ⁢                                                                                                          ⁢                                                      m                            2                            ∞                                                                                              )                                        ⁢                                          (                      ξ                      )                                                                            1                    +                                                                  m                        3                        ∞                                            ⁡                                              (                        ξ                        )                                                                                            .                                                                        (        33        )            If q has support in the ray (−∞,t1) then:
                              r          ⁡                      (            ξ            )                          =                                            (                                                m                                      11                    -                                                  +                                  ⅈ                  ⁢                                                                          ⁢                                      m                                          21                      -                                                                                  )                        ⁢                          (                              ξ                ;                t                            )                        ⁢                          ⅇ                                                -                  2                                ⁢                                                                  ⁢                ⅈ                ⁢                                                                  ⁢                ξ                ⁢                                                                  ⁢                t                                                          1            +                                          m                                  31                  -                                            ⁡                              (                                  ξ                  ;                  t                                )                                                                        (        34        )            is independent of t for t≧t1. It is also useful to observe that if r(ξ) is the reflection coefficient, determined by the potential q(t), then e−2iτξr(ξ) is the reflection coefficient determined by the time shifted potential qτ(t)=q (t−τ).
As m∞(ξ) is a unit vector valued function, the reflection coefficient r(ξ) uniquely determines m∞(ξ) and vice-versa Thus, the RF-pulse synthesis problem can be rephrased as the following inverse scattering problem:
Find a potential q(t) for the ZS-system so that the reflection coefficient r(ξ) satisfies Equation (33) for all real ξ.
The pulse synthesis problem makes no reference to the data connected with the bound states, i.e. {(ξj, Cj)}. Indeed these arefree parameters in the RF-pulse synthesis problem, making the problem highly underdetermined.
The flip angle profile, φ(ξ); is related to the scattering data by:
                              φ          ⁡                      (            ξ            )                          =                                            sin                              -                1                                      ⁡                          (                                                2                  ⁢                                                                                r                      ⁡                                              (                        ξ                        )                                                                                                                                  1                  +                                                                                                          r                        ⁡                                                  (                          ξ                          )                                                                                                            2                                                              )                                =                      2            ⁢                                                  ⁢                                                            sin                                      -                    1                                                  ⁡                                  (                                                                                b                      ⁡                                              (                        ξ                        )                                                                                                  )                                            .                                                          (        35        )            SLR pulses are usually designed using the flip angle profile, and the phase of the reflection coefficient of the designed pulse is determined indirectly. In this context, the reflection coefficient determined by the pulse envelope is an approximation to a function of the form eiφ(ξ)r(ξ) where r(ξ) is the “ideal” reflection coefficient. The function φ(ξ) is the phase error and it is implicitly determined by details of the SLR algorithm. If φ(ξ) is approximately linear over the support of r(ξ), then the magnetization can be approximately rephased. In this case, the actual rephasing time comes in part from t1, the maximum value of t within the effective support of q(t) (i.e., the largest interval where the value of the potential is large enough to produce a measurable effect), and in part from the phase factor, eiφ(ξ).
A formula for the energy of the pulse envelope may be stated in terms of the reduced scattering data. This formula is quite different from the formula taught by Pauly et al. that relates the energy in an SLR-pulse to a parameter, a0, which arises as an intermediate step in their construction. It does not provide an explicit formula for the energy in terms of the magnetization profile and auxiliary parameters. Moreover, it is not evident from this formula how the energy depends on the locations of the bound states. The underlying results from inverse scattering theory are due to Zakharov, Faddeev and Manakov.
Theorem 2. Suppose that q(t) is a sufficiently rapidly decaying potential for the ZS-system, with reflection coefficient r(ξ), and discrete data {(ξj,Cj), j=1, . . . , N}, then:
                                          ∫                          -              ∞                        ∞                    ⁢                                                                                      q                  ⁡                                      (                    t                    )                                                                              2                        ⁢                                                  ⁢                          ⅆ              t                                      =                                            1              π                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                log                  (                                      1                    +                                                                                                                    r                          ⁡                                                      (                            ξ                            )                                                                                                                      2                                                        ⁢                                                                          )                                ⁢                                  ⅆ                  ξ                                                              +                      4            ⁢                                          ∑                                  j                  =                  1                                N                            ⁢                              Im                ⁢                                                                  ⁢                                                      ξ                    j                                    .                                                                                        (        36        )            Prior Approaches to Pulse Synthesis
The oldest method of pulse synthesis is the Fourier transform method. The Fourier method is a simple linear approximation for this highly non-linear process. It provides usable results for small flip angles, but is largely an ad hoc method. Notwithstanding the fact that this method is an approximation for all non-zero flip angles, it gives surprisingly good results for flip angles up to about π/2. Over the past fifteen years two, slightly older, generally accepted method is the Shinnar-Le Roux- or SLR-algorithm introduced independently by M. Shinnar and his co-workers (U.S. Pat. No. 5,153,515; U.S. Pat. No. 5,572,126; Shinnar, et al., “The Synthesis of Pulse Sequences Yielding Arbitrary Magnetization Vectors,” Mag. Res. In Med., Vol. 12 (1989), pp. 74-88; and Shinnar et al. “The Application of Spinors to Pulse Synthesis and Analysis,” Mag. Res. in Med., Vol. 12 (1989) pp. 93-98) and P. Le Roux (U.S. Pat. No. 5,821,752; P. Le Roux, “Exact Synthesis of Radio Frequency Waveforms,” Proceedings of 7th Annual Meeting of SMRM, 1988, p. 1049; J. Pauly et al., “Parameter Relations for the Shinnar-Le Roux Selective Excitation Pulse Design Algorithm,” IEEE Trans. On Med. Imaging, Vol. 10 (1991), pp. 53-65) The second method is to use the inverse scattering transform, or IST, which was first introduced as a technique for pulse synthesis by Grunbaum et al. in “An Exploration of the Invertibility of the Bloch Transform,” Inverse Problems, Vol. 2 (1986), pp. 75-81, and later refined by J. Carlson, “Exact Solutions for Selective-Excitation Pulses,” J. of Mag. Res., Vol. 94 (1991), pp. 376-386, and “Exact Solutions for Selective-Excitation Pulses. II. Excitation Pulses with Phase Control,” J. of Mag. Res., Vol. 97 (1992), pp. 65-78; and Rourke et al., “The Inverse Scattering Transform and its Use in the Exact Inversion of the Bloch Equation for Noninteracting Spins,” J. of Mag. Res., Vol. 99 (1992), pp. 118-138. There are also approaches which use optimization techniques from control theory, but such approaches are not particularly relevant to the subject matter of the invention and thus will not be discussed further.
In the SLR-method, one approximates not the desired magnetization profile, but the flip angle profile derived from it. It is approximated by a rational function, r0 of eifΔ/2. Using standard polynomial design techniques, one finds a hard pulse, that is, a sequence of equally spaced δ-pulses:
            q      0        =                  ∑                  j          =          1                N            ⁢                        μ          j                ⁢                  δ          ⁡                      (                          t              -                              j                ⁢                                                                  ⁢                Δ                                      )                                ,which produces an approximation, m0, to the desired magnetization. Note, however, that only the flip angle is directly controlled. The phase of the excitation is determined implicitly by the algorithm used to construct r0. By specifying only the flip angle profile, the SLR approach retains more direct control over the total duration of the RF-pulse.
The forward scattering analysis has a well-defined limit for potentials of the form q0(t). In this case, the scattering matrix takes the special form:
                              s          ⁡                      (            ξ            )                          =                  [                                                                                          ⅇ                                          ⅈ                      ⁢                                                                                          ⁢                      N                      ⁢                                                                                          ⁢                      Δξ                                                        ⁢                                                            A                      0                                        ⁡                                          (                                              ⅇ                                                                              -                            ⅈ                                                    ⁢                                                                                                          ⁢                          Δ                          ⁢                                                                                                          ⁢                          ξ                                                                    )                                                                                                                                        ⅇ                                          ⅈ                      ⁢                                                                                          ⁢                      N                      ⁢                                                                                          ⁢                      Δξ                                                        ⁢                                                            B                      0                      *                                        ⁡                                          (                                              ⅇ                                                                              -                            ⅈ                                                    ⁢                                                                                                          ⁢                          Δ                          ⁢                                                                                                          ⁢                          ξ                                                                    )                                                                                                                                                                ⅇ                                                                  -                        ⅈ                                            ⁢                                                                                          ⁢                      N                      ⁢                                                                                          ⁢                      Δξ                                                        ⁢                                                            B                      0                                        ⁡                                          (                                              ⅇ                                                                              -                            ⅈ                                                    ⁢                                                                                                          ⁢                          Δ                          ⁢                                                                                                          ⁢                          ξ                                                                    )                                                                                                                                        -                                          ⅇ                                                                        -                          ⅈ                                                ⁢                                                                                                  ⁢                        N                        ⁢                                                                                                  ⁢                        Δξ                                                                              ⁢                                                            A                      0                      *                                        ⁡                                          (                                              ⅇ                                                                              -                            ⅈ                                                    ⁢                                                                                                          ⁢                          Δ                          ⁢                                                                                                          ⁢                          ξ                                                                    )                                                                                                    ]                                    (        37        )            where A0(z) and B0(z) are polynomials of degree N−1. The reflection coefficient r0(ξ), defined by q0(t), is the periodic function of period Δ−12π given by:
                                          r            0                    ⁡                      (            ξ            )                          =                                                            ⅇ                                                      -                    2                                    ⁢                  ⅈ                  ⁢                                                                          ⁢                  N                  ⁢                                                                          ⁢                  Δ                  ⁢                                                                          ⁢                  ξ                                            ⁢                                                B                  0                                ⁡                                  (                                      ⅇ                                                                  -                        ⅈ                                            ⁢                                                                                          ⁢                      Δξ                                                        )                                                                                    A                0                            ⁡                              (                                  ⅇ                                                            -                      j                                        ⁢                                                                                  ⁢                    Δξ                                                  )                                              .                                    (        38        )            
Thus, the SLR algorithm has two parts: 1. Find polynomials of the given degrees, so that r0(ξ) is, in a certain sense, an approximation to ri(ξ). 2. Use a recursive method for determining the coefficients, {μj}, so that q0(t) given as:
                                                        q              0                        ⁡                          (              t              )                                =                                    ∑                              j                =                1                            N                        ⁢                                          μ                j                            ⁢                              δ                ⁡                                  (                                      t                    -                                          j                      ⁢                                                                                          ⁢                      Δ                                                        )                                                                    ,                            (        39        )            has reflection coefficient r0(ξ).
Of course, a sum of δ-pulses is nonphysical, requiring infinite energy to realize. The RF-envelope that is actually used is a “softened” version of q0(t). For example, one could replace each ξjδ(t−jΔ) by a boxcar pulse of width Δ with the same area, leading to the softened pulse:
                                          q            1                    ⁡                      (            t            )                          =                              ∑                          j              =              1                        N                    ⁢                                                    μ                j                            Δ                        ⁢                                                            χ                                      (                                          0                      ,                      Δ                                        )                                                  ⁡                                  (                                      t                    -                                          j                      ⁢                                                                                          ⁢                      Δ                                                        )                                            .                                                          (        40        )            While the difference q0(t)−q1(t) can only be made small in the sense of generalized functions, the difference |r0(ξ)−r1(ξ)| can be made pointwise small, for ξ in a fixed interval, provided that none of the {μj} is too large. This is what is usually meant by the “hard pulse approximation” described by Shinnar et al.
Careful control of the duration is important in applications that involve imaging samples with a short T2, or spin-spin relaxation time. Spin-spin relaxation only becomes important once the magnetization has a significant transverse component. At least for minimum energy pulses, this occurs near the peak of the pulse. The duration of the pulse after the peak is almost equal to the rephasing time. Hence, in order to design pulses for experiments with a short T2, it is really only necessary to control the rephasing time, which is possible in IST pulse design. In practice, a pulse designed using the IST method has only slightly longer duration than the SLR pulse, with the same design parameters. In numerical simulations, an IST pulse appears to be slightly less susceptible to errors caused by spin-spin relaxation, than the comparable SLR pulse.
The SLR-method does not directly control the phase of the transverse magnetization profile nor do the “auxiliary parameters,” inherent in the pulse design problem, arise in this approach. In truth the SLR approach does not solve the pulse design problem, per se, though in many important instances, it produces a result which is adequate for most current applications.
From a practical perspective, both IST and SLR require considerable computation. From the inverse scattering formalism, it is clear that there are many possible RF-envelopes that will produce the same magnetization profile. There are also auxiliary parameters in the SLR method. Changing the auxiliary parameters that appear in SLR changes the phase of the magnetization profile, only leaving the flip angle profile unchanged. So changing the auxiliary parameters in SLR actually produces solutions to different pulse design problems, whereas varying the auxiliary parameters in IST produces different solutions to the same pulse design problem. In either case, some additional criteria are needed to determine how the auxiliary parameters should be chosen. The “real” problems of pulse synthesis are therefore:
(a) To choose the auxiliary parameters leading to an RF-envelope, “optimal” for a given application, which produces an approximation to a given magnetization profile; and
(b) To provide an algorithm that produces the optimal pulse.
Point (a) entails having an algorithm that can find the pulse, with a given magnetization profile, and an arbitrary specification of the auxiliary parameters. Prior to the work of the present inventors, this problem was unsolved in the MR literature. This “real” problem of pulse synthesis is solved herein and used to synthesize pulses of the type usable for MRI, for example.
In view of the above, there is a need in the art for a pulse generation algorithm that is as efficient as SLR though more flexible and that provides more direct control over the final result in terms of the input data